铁磁流体自由表面的局部径向斑图
Localised Radial Patterns on the Free Surface of a Ferrofluid
摘要：
本文研究了在均匀垂直磁场作用下铁磁流体表面局部轴对称（径向）斑图的存在性。我们正式调查所有可能的小振幅的解决方案，保持有界接近图案的中心（核心区域）和指数衰减远离图案的中心（远场区域）。结果提出了一个有限的深度，无限广阔的铁磁流体配备了线性磁化定律。这些模式在Rosensweig不稳定性处分叉，其中所施加的磁场强度达到临界阈值。建立了一个非自治PDE系统的局部化解决方案的技术，解决方案分解到一个独立的半径的基础上，减少了问题的非线性，非自治常微分方程的无限集。利用径向中心流形理论，分别在核心区和远场区构造了小振幅解的局部流形。最后，使用几何爆破坐标，我们匹配的核心和远场流形;任何解决方案，在于这个交叉点是一个局部的径向图案。三种不同的固定径向解决方案：A点和B点的解决方案，这是配备了两个不同的振幅标度律，并实现其最大振幅的核心，和环的解决方案，实现其最大振幅远离核心。这些解决方案完全对应于类的局部径向解决方案发现斯威夫特-霍恩伯格方程。不同的值的线性磁化强度和深度的铁磁流体的研究和参数区域中出现的各种局部径向解决方案被确定。本文所采取的方法概述了一条路线，严格建立在未来的轴对称局部图案的存在。
英文摘要：
This paper investigates the existence of localised axisymmetric (radial) patterns on the surface of a ferrofluid in the presence of a uniform vertical magnetic field. We formally investigate all possible small-amplitude solutions which remain bounded close to the pattern's centre (the core region) and decay exponentially away from the pattern's centre (the far-field region). The results are presented for a finite-depth, infinite expanse of ferrofluid equipped with a linear magnetisation law. These patterns bifurcate at the Rosensweig instability, where the applied magnetic field strength reaches a critical threshold. Techniques for finding localised solutions to a non-autonomous PDE system are established; solutions are decomposed onto a basis which is independent of the radius, reducing the problem to an infinite set of nonlinear, non-autonomous ODEs. Using radial centre manifold theory, local manifolds of small-amplitude solutions are constructed in the core and far-field regions, respectively. Finally, using geometric blow-up coordinates, we match the core and far-field manifolds; any solution that lies on this intersection is a localised radial pattern. Three distinct classes of stationary radial solutions are found: spot A and spot B solutions, which are equipped with two different amplitude scaling laws and achieve their maximum amplitudes at the core, and ring solutions, which achieve their maximum amplitudes away from the core. These solutions correspond exactly to the classes of localised radial solutions found for the Swift-Hohenberg equation. Different values of the linear magnetisation and depth of the ferrofluid are investigated and parameter regions in which the various localised radial solutions emerge are identified. The approach taken in this paper outlines a route to rigorously establishing the existence of axisymmetric localised patterns in the future.
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